Note that our notation here is mostly in-line with Ref.[1]} instead.
While \(A\) , particularly, as a frame of reference for position and
momentum observables, may have to be a system with some structure
under any practical consideration, so long as the transformation
considered is concerned, we only have to address its center of mass
degrees of freedom, which of course behaves exactly as those for
a single particle. Similarly for all, we may simply think about \(A\) , \(B\) ,
and \(C\)  each as a quantum particle. The expressions for the position and
momentum observables each has no components. Generalization to the
case that each is a three-vector of independent components would be
straightforward. The transformation as a quantum spatial translation
is easy to appreciate. The part of the position observables read as
classical ones would be exactly what one has in a classical theory.
The part of momentum observables is what is required to make the
full transformation a canonical one, i.e. to have the Poisson
bracket \(\frac{1}{i\hbar } [\cdot ,\cdot ]\)  or all \(\hat{x}\) -\(\hat{p}\) 
commutators preserved. Implicitly, the thinking about quantum
reference frame transformations has hidden in it an intuitive
but formally not so trivial [2]} picture of the position and
momentum observables as (noncommutative) coordinates of the
phase space for the quantum system. Quantum reference frame
transformations are symmetry transformations of the latter.
